Correct Option (d)
To determine the number of brown socks, we first define the variables and formulate the costs involved.
- Let the price of one pair of brown socks be x.
- Given that the price of a black pair was thrice that of a brown pair, the price of one pair of black socks is 3x.
- Let the unknown number of pairs of brown socks in the original order be y. The number of black socks in the original order is given as 5 pairs.
The correct bill, based on the original order, would be:
Original Bill = (Number of black pairs × Price of black pair) + (Number of brown pairs × Price of brown pair)
Original Bill = 5 * (3x) + y * x = 15x + xy
Due to the bill clerk's mistake, the number of black and brown pairs was interchanged. Thus, the mistaken bill was calculated for y black pairs and 5 brown pairs:
Mistaken Bill = (Number of black pairs in mistake × Price of black pair) + (Number of brown pairs in mistake × Price of brown pair)
Mistaken Bill = y * (3x) + 5 * x = 3xy + 5x
The problem states that the mistaken bill increased by 100% compared to the original bill. This implies the mistaken bill is twice the original bill:
Mistaken Bill = 2 × Original Bill
3xy + 5x = 2(15x + xy)
Now, we solve this equation for y:
3xy + 5x = 30x + 2xy
Subtract 2xy from both sides of the equation:
xy + 5x = 30x
Subtract 5x from both sides:
xy = 25x
Since x represents a price, it cannot be zero. Therefore, we can divide both sides by x:
y = 25
Thus, the number of pairs of brown socks in the original order was 25.
Incorrect Options:
Options 10, 15, and 20 are incorrect because substituting these values for y into the derived equation 3xy + 5x = 2(15x + xy) would not satisfy the condition that the mistaken bill is precisely 100% higher (i.e., double) than the correct bill. Only y = 25 maintains this proportional relationship between the correct and mistaken bills.